Maximal Subgroups in Finite Classical Groups

• Maximal subgroups in finite classical groups are defined as subgroups with odd index, characterized by precise geometric and arithmetic conditions.
• The classification employs Aschbacher’s framework, detailed 2-part combinatorial computations, and discriminant analysis to verify subgroup maximality.
• The results have significant implications for permutation group theory, design theory, and computational algebra by refining structural and algorithmic approaches.

Maximal subgroups in finite classical groups form a central object of paper in group theory, combinatorics, and computational algebra. They encapsulate both geometric and arithmetic structure, controlling group actions on natural modules, influencing permutation group theory, and dictating the possible overgroups of elementary subgroups. The classification of maximal subgroups of odd index for finite simple classical groups—such as $\mathrm{PSL}_n(q)$, $\mathrm{PSU}_n(q)$, $\mathrm{PSp}_n(q)$, and various orthogonal groups over fields of odd characteristic—synthesizes several decades of research and resolves subtle flaws in the early foundational work. The approach draws on Aschbacher’s program, discriminant computations, and number-theoretic analysis of indices, and has practical ramifications for design theory, permutation group theory, and related areas.

1. Structural Classification of Maximal Subgroups of Odd Index

A maximal subgroup $H$ of a finite simple classical group $G$ over a field $\mathbb{F}_q$ ($q$ odd) is said to have odd index if $|G : H|$ is odd. The paper establishes a precise, case-wise classification for the following families:

• $G = \mathrm{PSL}_n(q)$, $n \geq 2$
• $G = \mathrm{PSU}_n(q)$, $n \geq 3$
• $G = \mathrm{PSp}_n(q)$, even $n \geq 4$
• $G = \mathrm{P}\Omega_n(q)$, odd $n \geq 7$
• $G = \mathrm{P}\Omega_n^{(\pm)}(q)$, even $n \geq 8$

Maximal subgroups $H$ of odd index are divided into 23 explicit cases. The most significant categories include:

• Automorphism centralizers: Centralizers of a field automorphism of odd prime order.
• Stabilizers of subspaces: Maximal stabilizers of $m$-dimensional subspaces $U \leq V$ (projective module), with the condition $n \succcurlyeq m$ (in binary expansions; i.e., $m$ does not exceed $n$).
• Decomposition stabilizers: Stabilizers of orthogonal or totally singular decompositions $V = V_1 \oplus \cdots \oplus V_k$, with constraints on $m$, $k$, and the subspace discriminants.
• Exceptional isomorphism types: Specific small-dimensional cases that are only realized for select parameters ($n$, $q$).

The subgroup type, index, and necessary parameter conditions (e.g., congruence restrictions on $q$, discriminant formulas) are precisely dictated in each case.

2. Correction, Revision, and Consistency

Earlier work—particularly that based on Kleidman’s thesis [9]—contained critical flaws in at least three cases:

• Case (6): For decomposition stabilizers in $\mathrm{PSL}_n(q)$, corrections adjust the permitted $q$ for $m=1$ (requiring $q \equiv 1 \mod 4$ and excluding certain low-dimensional exceptions).
• Case (10): For orthogonal groups with $D(V)=1$, new restrictions on the sign and arithmetic properties of subspaces (including $(m, q) \neq (3,3)$) are provided.
• Case (21): For $\mathrm{PSp}_4(q)$, refined congruence conditions on $q$ ($q \equiv \pm 1, \pm 3 \mod 8$) ensure the index is odd.

The revised theorem confirms that the permitted $H$ in each of the 23 cases are genuinely maximal and that their indices have the correct parity. This reconciliation guarantees both completeness and accuracy when compared to Aschbacher’s framework and the tables of known subgroups in low-dimensional cases.

3. Methodological Approach and Key Theorems

The classification leverages four principal pillars:

• Finite simple group classification: All structural results are rooted in the global classification of finite simple groups and their automorphism groups.
• Aschbacher’s subdivision: Subgroups are systematically analyzed by the eight classical Aschbacher families ($C_1$–$C_8$), allowing for identification of geometric versus algebraic types.
• Combinatorial 2-part computations: Determination that $|G : H|_2 = 1$ (the maximal power of 2 dividing $|G : H|$) is central. Detailed combinatorial and arithmetic verifications appear throughout, using explicit formulas for the order of $G$ and $H$.
• Quadratic form invariants: Subgroup maximality often relies on structural properties of the natural module, especially discriminant $D(V)$ and sign $\operatorname{sgn}(V)$. The key formulas include:
  • Subspace decomposition: $V = U \oplus U^\perp$
  • Discriminant product: $D(V) = D(U) \cdot D(U^\perp)$
  • Sign product: $$\operatorname{sgn}(V) = \operatorname{sgn}(U) \cdot \operatorname{sgn}(U^\perp)$$

Verification involves both low-dimensional explicit tables and recursive techniques for higher dimensions.

4. Implications and Applications

The exhaustive nature of the classification resolves long-standing questions in both group theory and applications:

• Primitive permutation groups: The result directly determines the primitive permutation groups with odd degree, informing the structure of rank 3 permutation groups and related combinatorial objects.
• Computational group theory: Algorithms for group actions, maximal subgroup search, and subgroup conjugacy leverage these case-wise results to optimize performance and avoid erroneous classification in odd index settings.
• Further research: The description enables investigations into automorphism groups of combinatorial designs, stabilizers of geometric structures, and computational enumeration of maximal subgroups in classical groups.
• Design theory and structure analysis: Applications extend to the synthesis of designs, automorphism group classification for geometries, and probabilistic generation in finite groups.

5. Technical Formulations and Explicit Conditions

The following table summarizes several of the crucial formulas and their roles:

Formula/Condition	Role	Context
$V = U \oplus U^\perp$	Module decomposition	Quadratic/symplectic/orthogonal groups
$D(V) = D(U) \cdot D(U^\perp)$	Discriminant splitting	Orthogonal group index analysis
$|G:H|_2 = 1$	Odd index criterion	Verification across cases
$n \succcurlyeq m$ (binary)	Subspace stabilizer	Linear/unitary/symplectic groups
$(q-1)m/4$ even/odd	Discriminant computation	Classifies non-degenerate subspaces

Additional congruence and discriminant conditions appear in each explicit case for verification.

6. Summary and Outlook

The complete case-by-case theorem for maximal subgroups of odd index in finite simple classical groups over odd characteristic fields, incorporating the necessary corrections to earlier work, establishes the structural framework for all such subgroups. It unifies geometric, arithmetic, and representation-theoretic methods, anchored in Aschbacher’s subdivision and meticulous 2-part index analysis. The implications span recognition algorithms, design theory, computation, and further theoretical developments in permutation groups and subgroup structure.

This classification forms a definitive reference point for subsequent research and applications involving subgroup maximality, primitive group actions, and the geometric structure of classical groups. The interplay between arithmetic criteria (such as binary expansion constraints, discriminant products, and congruence conditions) and geometric stabilizer properties illustrates the depth and rigidity of subgroup structure in finite classical groups.